A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 5, 1, 1, 2, 1, 1, 1

Offset

1,4

Comments

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Links

Table of n, a(n) for n=1..87.

Example

The a(n) factorizations for n = 36, 360, 720, 192, 288:
  (36)     (360)    (720)     (192)      (288)
  (6*6)    (5*72)   (8*90)    (3*64)     (8*36)
  (2*2*9)  (8*45)   (9*80)    (4*48)     (9*32)
  (3*3*4)  (9*40)   (10*72)   (6*32)     (16*18)
           (10*36)  (16*45)   (12*16)    (2*144)
           (5*8*9)  (5*144)   (3*8*8)    (6*6*8)
                    (5*9*16)  (4*6*8)    (2*2*72)
                    (8*9*10)  (3*4*16)   (2*9*16)
                              (3*4*4*4)  (3*3*32)
                                         (2*2*8*9)
                                         (3*3*4*8)
                                         (2*2*2*36)
                                         (2*2*2*2*2*9)

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
prisig[n_]:=If[n==1, {}, Last/@FactorInteger[n]];
Table[Length[Select[facs[n], stableQ[#, Intersection[prisig[#1], prisig[#2]]!={}&]&]], {n, 100}]

Crossrefs

A001055 counts factorizations.

A118914 is sorted prime signature.

A124010 is prime signature.

A336737 counts factorizations with intersecting signatures.

Cf. A000372, A003182, A006126, A109298, A112798, A293606, A294068, A305844, A321469, A336424.

Sequence in context: A227350 A107345 A358780 * A000688 A295879 A322453

Adjacent sequences: A336733 A336734 A336735 * A336737 A336738 A336739

Keyword

nonn

Author

Gus Wiseman, Aug 06 2020

Status

approved